Generalized Fractional Heat Conduction in a One-Dimensional Functionally Graded Material Layer

Abstract

Functionally graded materials (FGMs) have been widely used. This work studies a heat conduction problem in an FGM layer with different exponential gradients. Based on a generalized fractional heat conduction theory with phase lag of heat flux, a mixed initial-boundary value problem is solved. Analytical expression for temperature change in the Laplace transform domain is derived. Numerical results of the transient temperature response in the time domain are evaluated by applying a numerical inversion of the Laplace transform. Two representative boundary conditions related to given temperature or heat convective transfer are discussed. For different Biot numbers, the effects of phase lag of heat flux, fractional order, and material properties on temperature field are illustrated graphically. A comparison of the temperature fields based on the non-Fourier model and classic Fourier model is made. The obtained results show that wave-like behaviors may occur for the generalized fractional heat transfer, which does not occur for the classical Fourier heat conduction. The material properties play an important role in the heat transfer process. The derived results are of benefit to the design of materials for thermoresistance and abstraction of heat.